Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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1
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3answers
22 views

Show $\frac{(2n)!}{n!\cdot 2^n}$ is an integer for $n$ greater than or equal to $0$

Show $$\frac{(2n)!}{n!\cdot 2^n}$$ is an integer for $n$ greater than or equal to $0$. Could anyone please help me with this proving? Thanks!
0
votes
2answers
32 views

Prove (or derive) the de Polignac formula for the prime decomposition of $n!$

I can't seem to find any papers published dedicated to show that the de Polignac formula has a rigorous derivation. From Wikipedia's entry for the formula: Let $n \geq 1$ be an integer. The prime ...
0
votes
1answer
30 views

Ratios between Factorial numbers and the sum of their factors

Let a factorial number be called $f!$. Let the sum of its factors be called $S(f!)$. Let the ratio between the two be “r”, such that $r=\frac{S}{f!}$. It is conjectured that: ...
-2
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0answers
33 views

Is there a function that shows n!-lcm(1,2,3…n)? [on hold]

1!-lcm(1)=0 2!-lcm(1,2)=0 3!-lcm(1,2,3)=0 4!-lcm(1,2,3,4)=12 5!-lcm(1,2,3,4,5)=60 I have not looked into this incredibly deeply, but I found that there may be some sort of connection with primes, ...
4
votes
0answers
43 views

What steps have been taken so far to solve Brocard's Problem?

The equation is $$n!+1=m^2$$ where $n$ and $m$ are natural numbers. Brocard's Problem asks whether there are solutions for n other than $4, 5, 7$. The only improvement I have found that people have ...
1
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0answers
22 views

Is there a function of, say, x and y that would take the first x factors in a factorial and return a xCy amounts of terms with y factors in each term?

What I'm basically looking for is described in the title. Here are some examples of what the function I'm looking for should do. Is there an existing function that does this? Even if not, are there ...
3
votes
1answer
34 views

Split Factorial of n

How can I split integers up to n into two groups such that the difference of the product of each group is as low as possible? Is there a way to optimize the selection for each group in order to ensure ...
2
votes
2answers
44 views

What is the logic/theorem/derivation behind finding the exponent of p in n! By [n/p] + [n/p^2] + [n/p^3] + …? [duplicate]

The exponent of prime number of 3 in 100! is 48. It means 100! is divisible by $3^48$ $$E_3(100!) = \left\lfloor\frac{100}3\right\rfloor + \left\lfloor\frac{100}{3^2}\right\rfloor + ...
1
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2answers
62 views

Any shortcut to calculate factorial of a number (Without calculator or n to 1)?

I've been searching the internet for quite a while now to find anything useful that could help me to figure out how to calculate factorial of a certain number without using calculator but no luck ...
6
votes
3answers
423 views

Some trouble with the induction

Prove, that for any positive integer $n \geqslant 2$ we have the inequality $$ \frac{ 4^n }{ n+1 } < \frac{ (2n)! }{ (n!)^2 }.$$ For $n=2$ the inequality is true. Directly just take and ...
-1
votes
1answer
11 views

Common factors /greatest common factors /least common factor

See I am baffling with the concept of above mentioned title I saw one question they wrote find common factors /gcf/lcm of 42,294,882 Please help me out in getting the concept Thanks
0
votes
1answer
37 views

How do I prove the formula for multichoose?

In combinatorics, there is a formula "$n$ multichoose $k$", which is the way of making a multiset having $k$ elements choosing out of $n$ options. "$n$ multichoose $k$" is the same as "$(n+k-1)$ ...
0
votes
0answers
36 views

How to write $n!=a^{\alpha_0}(a+1)^{\alpha_1}(a+2)^{\alpha_2}\cdots(a+r)^{\alpha_r}$?

We know that $n!=n(n-1)(n-2)\cdots3\cdot2\cdot1, n\in \mathbb N$. Now I am willing to write $n!$ as $a^{\alpha_0}(a+1)^{\alpha_1}(a+2)^{\alpha_2}\cdots(a+r)^{\alpha_r}$ where $a, r, \alpha\in \mathbb ...
1
vote
1answer
51 views

Approximating $\frac{(kn)!}{(n!)^k}$

Is there any approximations for the form $$\frac{(kn)!}{(n!)^k},$$ where $n$ and $k$ are positive integers? $n$ is not necessary much larger than $k$?
0
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3answers
56 views

How do I derive $n!$ from this series?

I am reading a book where the following reduction is performed, but it's not explained exactly what is going on. I'm sorry if this is a dumb question, but I simply don't get how we are deriving the ...
8
votes
4answers
143 views

Is there an integral representation for $\frac{1}{n!}$?

I know that $n!$ has various integral representations, for instance the $\Gamma$ function. I was wondering if $\frac{1}{n!}$ has an integral representation.
1
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2answers
44 views

How do I evaluate this $ \lim \frac{\zeta(n)} {({n)!}} , n\to\infty $ if it was existed?

Is there someone who can show me how do I evaluate this limit $$ \lim_{n\to +\infty} \frac{\zeta(n)} {n!} $$ if it exists ? Thank you for any help.
6
votes
0answers
74 views

Could this approximation be made simpler ? Solve $n!=a^n 10^k$

I need to find the smallest value of $n$ such that $$\frac{a^n}{n!}\leq 10^{-k}$$ in which $a$ and $k$ are given (these can be large numbers). I set the problem as : solve for $n$ the equation ...
6
votes
1answer
88 views

Why is $!0 = 1$?

The subfactorial function is defined as: $$!n = n!\sum_{i=0}^n\dfrac {(-1)^i} {i!}$$ I was curious and wanted to find out what $!0$ came out to be. Since I couldn't use it in the sum above, I used a ...
0
votes
2answers
47 views

Simplifying expressions with factorials

If $$\large{a_n = \frac{x^n}{2^n n!}}$$ , Then find $$\large{ \frac{a_{n +1}}{a_n}}$$ .. I tried the following: $$\large{a_{n + 1} = \frac{x^n}{2^n n!} + \frac{2^n n!}{ 2^n n!} = \frac{x^n + 2^n ...
4
votes
3answers
106 views

How to find $\lim _{ n\to \infty } \frac { ({ n!) }^{ 1/n } }{ n } $? [duplicate]

How to find $\lim _{ n\to \infty } \frac { ({ n!) }^{ 1/n } }{ n } $ ? I tried taking using logarithm to bring the expression to sum form and then tried L Hospital's Rule.But its not working.Please ...
1
vote
2answers
61 views

How do I find the probability of some elements being together inside a randomly arranged set?

If I have a total of $n$ balls made of $k$ red balls and $(n-k)$ green balls and I arrange them all randomly in a line, how can I calculate the probability $x$ of a group of $y$ red balls being ...
2
votes
1answer
66 views

Why do the closest primes whose distance $d \gt 1$ to $c(n)=\frac{(n+1)!+n!}{2}$ have always $d \in \Bbb P$?

I have made the following observation: define the center of $n!$ and $(n+1)!$, $c(n)$, as the number located exactly in the middle of $(n+1)!$ and $n!$. Def: $\forall n \gt 2\ , \ ...
1
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1answer
34 views

Proving that $\Delta x^{(n)} = n x^{(n-1)}$

Define $\Delta f(x) = f(x+1) - f(x)$ (the difference operator). Define $x^{(n)} = x(x-1) \dots (x-n+1)$ (the falling factorial function). There's a rather simple theorem which shows that $\Delta ...
0
votes
1answer
41 views

What function does Wolfram Alpha plot instead of the factorial?

Look to the second graph where Wolfram Alpha gives a continuous factorial function: What is the second graph? It is not the gamma function, since that has $\Gamma(-1)=0!=1$.
-3
votes
1answer
88 views

Solve this equation : $(2x)! = (x)! (x+2)!$ [closed]

Solve this equation : $(2x)! = (x)! (x+2)!$
10
votes
6answers
253 views

Determine whether $\frac{1000!}{100!^{10}}$ is an integer

Can you give an idea, how to find out whether the result of ${1000!}/{100!^{10}}$ an integer. Modulo division? But what I met was about powers like $2^{100}/125$...
9
votes
2answers
73 views

When is a binomial coefficient a factorial, i.e. when is $\binom{m}{j} = n!$ for some $n,m,j$?

As stated in the title: when is a binomial coefficient a factorial, i.e. when is $\binom{m}{j} = n!$ for some $m,j,n$? I was thinking about this problem a couple of days ago because in all my years of ...
1
vote
1answer
58 views

Number of Divisors of N factorial

Say d(N) = Number of factors of N! Briefly: I wish to know if there is a Recurrence relation for this problem Now I wish to Know if there is a way to calculate d(N) in terms of previously calculated ...
7
votes
7answers
176 views

Prove $\lim\limits_{n\to\infty}\frac{1}{\sqrt[n]{n!}}=0$

I used $$(n!)^{\frac{1}{n}}=e^{\frac{1}{n}\ln(n!)}=e^{\lim\limits_{n\to\infty}\frac{1}{n}\ln(n!)}$$ Then using Stirling's approximation and L'Hospital's rule on ...
1
vote
1answer
80 views

Solving a little Diophantine equation:$(n-1)!+1=n^m$ [duplicate]

How can I solve this Diophantine equation: $$(n-1)!+1=n^m$$ with $n,m$ positive integers? From Wilson's theorem we can note that $n$ is a prime number. I proved to rewriting the equation ...
7
votes
2answers
145 views

Show that $\lim\limits_{n \to \infty} \frac{(n!)^{1/n}}{n}= \frac{1}{e}$ [duplicate]

Show that $$\lim_{n \to \infty} \left\{\frac{(n!)^{1/n}}{n}\right\} = \frac{1}{e}$$ What I did is to let $U_n = \dfrac{(n!)^{\frac{1}{n}}}{n}$ and $U_{n+1} = ...
0
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1answer
24 views

Find number of possible combinations from list of items [closed]

I have the following list: Product Icecream Banana Strawberry Vanilla The possible combinations (based on the spec) are 5: ...
2
votes
1answer
48 views

Total possible ways of representing n! as a sum of two or more consecutive positive integers.

I need to calculate total possible ways of representing $n!$ as a sum of two or more consecutive positive integers. Example : $3!=1*2*3=6$ and $6=1+2+3$ the only one possible way. Answer : $1$ The ...
0
votes
3answers
31 views

Permutations and Combinations equation [closed]

I cannot figure out how I am supposed to start in solving for $n$ in this equation. $$_nP_4=84(_nC_2)$$ Thank you!
2
votes
2answers
90 views

Total number of divisors of factorial of a number

I came across a problem of how to calculate total number of divisors of factorial of a number. I know that total number of divisor of a number $n= p_1^a p_2^b p_3^c $ is $(a+1)*(b+1)*(c+1)$ where ...
1
vote
2answers
41 views

Find all natural numbers $m,n$ which :$m!+n!+10$ is perfect cube?

I would be interest to invesitigate for all natural numbers $m,n$ which: $m!+n!+10$ is perfect cube ?
0
votes
0answers
32 views

estimations in the birthday paradox?

The birthday paradox is the famous following problem: What is the probability $p_n$ that at least $2$ persons amongst $n$ persons chosen at random have the same birthday? Leap years are not taken ...
1
vote
3answers
60 views

The derivative of $x!$ and its continuity

is the factorial of fractions and negative numbers defined? If yes, then what is its graph? Also please find its domain. Our teacher said the factorial of a fraction is the fraction itself. He also ...
6
votes
0answers
101 views

Partial sums of falling factorials

I want to know if there exists some way, approximate or exact, to do a partial sum of falling factorials of the kind: $$\sum_{k=i}^{n}(a+k)_{h}$$ where all are constants. And I'm interested too in ...
0
votes
0answers
45 views

Is there an easy way to calculate the elementary symmetric functions?

Hello I am interested in the question of what, generally, is the sum of the series of reciprocals of a series of numbers we know its sum. I have particular interest in the Zeta function, which I ...
0
votes
4answers
54 views

Prove that $n! \geq {\lfloor n/2 \rfloor}^{\lfloor n/2 \rfloor}$

I am trying to see why it holds that, for $n \in \mathbb{N},$ $$n! \geq {\lfloor n/2 \rfloor}^{\lfloor n/2 \rfloor}.$$ I would appreciate help to see this.
2
votes
3answers
61 views

Why is this equality involving factorials true?

$$ (n +1)! -1 +(n +1)(n +1)! = (n +2)! -1 $$ Can someone explain me how in the world is this true? :D Thanks (yes I'm trying to understand induction).
5
votes
1answer
118 views

Prove by combinatorial method that $ \frac{(2m)! \cdot (2n)!}{(m)! \cdot (n)! \cdot (m+n)!} $ is an integer [duplicate]

Prove that $$ \dfrac{(2m)! \cdot (2n)!}{(m)! \cdot (n)! \cdot (m+n)!} $$ is a positive integer, where $(m,n) \in \mathbb{Z^{+}}$ I have already solved it using Legendre's Formula ...
2
votes
4answers
109 views

How many zeros are there in the number $50!$? [duplicate]

How many zeros are there in the number $50!$? My attempt: The zeros in every number come from the 10s that make up the number. The 10s are, in turn, made up of 2s and 5s. So: $\frac{50}{5*2} = 5$ ...
0
votes
1answer
35 views

Maximizing the following function

I need to find values of $k_1$, $k_2$ and $k_3$ that maximize $C^{m_1}_{mm_1} \cdot C^{m_2}_{k_1-mm_1} \cdot C^{n_1}_{nn_1} \cdot C^{n_2}_{k_2-nn_1} \cdot C^{p_1}_{pp_1} \cdot C^{p_2}_{k_3-pp_1}$ ...
6
votes
2answers
144 views

Proof of an identity of $n!$

I came up (numerically) with an identity concerning n! and I was wondering about a proof of it. Here it is: \begin{align} \ n! &= \sum_{r=0}^{n} { \binom{n}{r} (-1)^r(k-r)^n } \quad \forall n ...
0
votes
0answers
44 views

Modular arithmetic, specifically n! mod m

Is there a theorem that makes solving $$ n! \equiv x \mod m $$ knowing that both $n$ and $m$ are prime? And if not, what would be the best way to go about finding $x$? cheers
1
vote
3answers
85 views

Trying to show that $\sum_{1}^{\infty} \frac{n^n}{n!} $ diverges.

I have been trying to prove this using the ratio test $|\frac{A_{n+1}}{A_n}|$ , which leads me to this expression: $$\left|\frac{(n+1)^{n+1}}{(n+1)!}\cdot ...
1
vote
3answers
71 views

Calculate $\lim_{n\to\infty}\frac{5n^n}{3n!+3^n}$

Could I please have a hint for finding the following limit?$$\lim_{n\to\infty}\frac{5n^n}{3n!+3^n}$$